The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators

Abstract

This paper deals with the static Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$ The system is reformulated in quaternion analysis by Kravchenko in the form $$\mathcal {L}F=0$$ with $$\mathcal {L}F=DF+F\alpha $$ . We consider special cases of the coefficient function $$\Phi =\Phi _0(x_0)\Phi _1(x_1)\Phi _2(x_2)$$ and prove the existence of four generalized Cauchy kernels of the operator $$\mathcal {L}$$ . We construct four explicit generalized Cauchy kernels in the case $$\Phi =x_0^{2p}x_1^{2m}x_2^{2n}$$ .